Small-strain elastic response

At temperatures sufficiently below the glass transition and under stresses well below the plastic yield stress to be defined later, all polymers exhibit reversible elastic behavior, which is quite often anisotropic, particularly when it relates to a polymer product that has undergone substantial prior deformation processing. 

We first develop the generalized Hooke s law of energy elasticity as the linear connection between stress and strain in tensorial form and then proceed to consider the most relevant special forms for cases of high material symmetry. 

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

Finally, we consider theoretical or computational means of determining the ideal elastic constants of some polymers together with their temperatme dependences and compare these with experimental values, determined, as much as possible, under conditions that are free of viscoelastic relaxations. We then provide small-strain energy-elastic constants of a variety of both glassy and semi-crystalline polymers. 

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The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

The mechanical concepts of stress are outlined in Fig. 1, with the axes reversed from that employed by mechanical engineers. The three salient features of a stress-strain response curve are shown in Fig. la. Initial increases in stress cause small strains but beyond a threshold, the yield stress, increasing stress causes ever increasing strains until the ultimate stress, at which point fracture occurs. The concept of the yield stress is more clearly realised when material is subjected to a stress and then relaxed to zero stress (Fig. Ih). In this case a strain is developed but is reversed perfectly - elastically - to zero strain at zero stress. In contrast, when the applied stress exceeds the yield stress (Fig. Ic) and the stress relaxes to zero, the strain does not return to zero. The material has irreversibly -plastically - extended. The extent of this plastic strain defines the residual strain. 

To this point, we have limited the discussion to small strains—that is, small deviations from the equilibrium bond distance, such that all imposed deformations are completely recoverable. This is the elastic response region, one that virtually all materials possess (see Figure 5.8). What happens at larger deformations, however, is dependent to some... 

Assuming elastic response of the brittle film, the critical strain e which induces the cracking of the film can be deduced from relation. Typical material properties used in our calculations are given in Table 2. As discussed earlier, the residual strains are small relative to the critical strains, and are neglected in this analysis. The calculated critical cracking strains... 

In practice, interfaces are often subjected to a combination of the deformations mentioned. As in bulk rheology, there are some other variables. First, the response of a material to a force can be elastic or viscous. Elastic response means immediate deformation, where the strain (relative deformation, i.e., tan a. in shear and AA/A in dilatation) is related to the force on release of the force, the strain immediately becomes zero. In viscous deformation, the force causes flow or, more precisely, a strain rate (d tan a/dt or d In Ajdty, this occurs as long as the force lasts, and upon release of the force the strain achieved remains. For most systems, the behavior is viscoelastic. Second, deformation can be fast or slow, and time scales between a microsecond and more than a day may be of importance. Third, the relative deformation (strain) applied can be small—i.e., remain close to the equilibrium situation—or be large. 

A rubber-like solid is unique in that its physical properties resemble those of solids, liquids, and gases in various respects. It is solidlike in that it maintains dimensional stability, and its elastic response at small strains (<5%) is essentially Hookean. It behaves like a liquid because its coefficient of thermal expansion and isothermal compressibility are of the same order of magnitude as those of liquids. The implication of this is that the intermolecular forces in an elastomer are similar to those in liquids. It resembles gases in the sense that the stress in a deformed elastomer increases with increasing temperature, much as the pressure in a compressed gas increases with increasing temperature. This gas-like behavior was, in fact, what first provided the hint that rubbery stresses are entropic in origin. 

In classical terms, the mechanical properties of elastic solids can be described by Hooke s law, which states that an applied stress is proportional to the resultant strain but is independent of the rate of strain. For liquids, the corresponding statement is known as Newton s law, with the stress now independent of the strain but proportional to the rate of strain. Both are hmiting laws, valid only for small strains or rates of strain, and although it is essential that conditions involving large stresses, leading to eventual mechanical failure, be smdied, it is also important to examine the response to small mechanical stresses. Both laws can prove useful under these circumstances. 

Fig. 12.9 A simple, rigid-plastic-hardening representation of the constitutive response of a material, given by eq. (12.29), accommodating an early elastic-range modification at small strain.

Even in the apparently linear range, the response to stress should be considered as viscoelastic rather than elastic. Most polymers that behave in a linear, viscoelastic manner at small strains (< 1 %) behave in a nonlinear fashion at strains of the order of 1 % or more. However, in a fibrous composite, the resin may behave quite differently than it would in bulk. Stress and strain concentrations may exceed the limiting values for linearity in localized regions. Thus the composite may exhibit nonlinearity (Ashton, 1969 Trachte and DiBenedetto, 1968), as is the case with particulate-filled polymers (Section 12.1.2). Although nonlinearity at low strains is characteristic, Halpin and Pagano (1969) have predicted constitutive relations for isotropic linear viscoelastic systems, and verified their prediction using specimens of fiber-reinforced rubbers. 

A very small stress a working on the polymer melt will reveal the elastic response of the fluid, and the strain... 

The experiment we introduced at the beginning of the previous subsection is also called the creep experiment. A small stress of Gq is imposed on a solid sample for a time period of to at a constant temperature after the stop of stress, the strain of changing with the time period of t monitors the relaxatirMi curve. There are four typical responses separately corresponding to viscous, elastic, anelastic and viscoelastic responses, as illustrated in Fig. 6.8. The creep curve of polymer viscoelasticity exhibits both instant and retarded elastic responses upon imposing and removal of the stress, and eventually reaches the permanent deformation. 

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